相对论性超流体中BCS-BEC转化
摘要
我们构建了相对论性的玻色费米模型以研究超流体的BEC-BCS转变(BEC是玻色爱因斯坦凝聚的缩写,BCS是Bardeen-Copper-Schrieffer的缩写).模型不仅包含基本组元费米子,也包含能反映库珀对玻色子属性的玻色自由度.在平均场的近似下,我们可以忽略热玻色子和费米子的耦合,而通过调节玻色子质量和化学势的差别实现超流体的BEC-BCS转变.通过这个模型,我们也能得到凝聚和非凝聚玻色子以及配对和非配对费米子的粒子数随转变参数的变化规律.我们还发现在强耦合区域,系统中出现反粒子,这是相对论性BEC-BCS转变的特征.作为模型的应用,我们首次研究了非对称费米子配对的BEC-BCS转变,发现在强耦合的一些参数区域原来的非对称费米子配对稳定性消失.
更进一步,我们研究了BEC-BCS转变中的涨落效应.我们利用CJT(Cornvwall-Jaciw-Tomboulis的缩写)公式体系引入高阶相互作用的贡献从而构建了超越平均场近似的理论描述.玻色子的非零振荡模式提供了配对的量子涨落,其与费米子的相互作用给出2PI(两粒子不可约图)的有效势.这个有效势在BEC区域将改变系统的性质.我们计算了超流凝聚量、相变温度,以及各组分粒子所占比例随玻色子质量平方的变化.同时,我们还计算了玻色子的解离温度,在此温度之上,玻色子开始衰变.我们在BEC端找到了解离温度大于相变温度的参数区域,表明在此区域即使在相变温度之上玻色子也可以真实存在.
我们的这个相对论性的玻色费米模型可以用来研究致密物质中色超导,而且可以推广用来讨论核物质和夸克物质在低温高密条件下的连续演化.
本论文的创新之处如下:
·本文首次用玻色费米模型研究了相对论性超流体中的BCS-BEC转化.
·在平均场近似框架下我们得到了凝聚和非凝聚玻色子以及配对和非配对费米子的粒子数随转变参数的变化规律.
·我们发现在强耦合区域系统中出现反粒子,这是相对论性BEC-BCS转变的特征.
·我们首次研究了非对称费米子配对的BEC-BCS转变,发现在强耦合的一些参数区域原来的非对称费米子配对稳定性消失.
·我们还研究了BEC-BCS转变中的涨落效应对转化的影响,并找到稳定玻色子存在的参数区域,其标志为解离温度大于相变温度.
We investigate the crossover from Bardeen-Cooper-Schrieffer(BCS)pairing to a Bose-Einstein condensate(BEC)in a relativistic superfluid within a boson-fermion model.The model includes,besides the fermions,separate bosonic degrees of free-dom,accounting for the bosonic nature of the Cooper pairs.The crossover is realized by tuning the difference between the boson mass and boson chemical potential or directly the bare boson mass as a free parameter.As a mean field approxima-tion(MFA),we neglect the coupling between the thermal bosons with the fermions, the model yields populations of condensed and uncondensed bosons as well as gapped and ungapped fermions throughout the crossover region for arbitrary temperatures. Moreover,we observe the appearance of antiparticles for sufficiently large values of the crossover parameter.As an application,we study pairing of fermions with im-balanced populations and find the disappearing of the stable solution in the strong coupling region.
To go beyond the MFA we use Cornwall-Jackiw-Tomboulis(CJT)formalism to include higher order contributions.The quantum fluctuations of the pairing condensate is provided by bosons in non-zero modes,whose interaction with fermions gives the two-particle-irreducible(2PI)effective potential.It changes the crossover property in the BEC regime.We calculate the condensate,the critical temperature T_c and particle abundances as functions of crossover parameter the squared boson mass.We also compute the dissociation temperature T* for bosons,above which they are not stable and start to decay.We find the crossover parameter range for T~*＞T_c,which indicates the existence of bosons even above T_c but below T~*.
The model can potentially be applied to color superconductivity in dense quark matter at strong couplings.Or it can be extended to discuss the quarkoyinic conti-nuity.
The main innovations in this thesis are listed below:
●For the first time,we investigate the crossover from BCS pairing to a BEC in a relativistic superfluid within a boson-fermion model.
●In mean field approximation,the model yields populations of condensed and uncondensed bosons as well as gapped and ungapped fermions throughout the crossover.
●We observe the appearance of antiparticles for sufficiently large coupling as a characteristic of relativistic model.
●We study the pairing of fermions with imbalanced populations in BCS-BEC crossover,and find the disappearing of the stable solution in the strong coupling region.
●We also study the fluctuation effect in the crossover and find the parameter region for stable bosons where T~*＞T_c.
更进一步,我们研究了BEC-BCS转变中的涨落效应.我们利用CJT(Cornvwall-Jaciw-Tomboulis的缩写)公式体系引入高阶相互作用的贡献从而构建了超越平均场近似的理论描述.玻色子的非零振荡模式提供了配对的量子涨落,其与费米子的相互作用给出2PI(两粒子不可约图)的有效势.这个有效势在BEC区域将改变系统的性质.我们计算了超流凝聚量、相变温度,以及各组分粒子所占比例随玻色子质量平方的变化.同时,我们还计算了玻色子的解离温度,在此温度之上,玻色子开始衰变.我们在BEC端找到了解离温度大于相变温度的参数区域,表明在此区域即使在相变温度之上玻色子也可以真实存在.
我们的这个相对论性的玻色费米模型可以用来研究致密物质中色超导,而且可以推广用来讨论核物质和夸克物质在低温高密条件下的连续演化.
本论文的创新之处如下:
·本文首次用玻色费米模型研究了相对论性超流体中的BCS-BEC转化.
·在平均场近似框架下我们得到了凝聚和非凝聚玻色子以及配对和非配对费米子的粒子数随转变参数的变化规律.
·我们发现在强耦合区域系统中出现反粒子,这是相对论性BEC-BCS转变的特征.
·我们首次研究了非对称费米子配对的BEC-BCS转变,发现在强耦合的一些参数区域原来的非对称费米子配对稳定性消失.
·我们还研究了BEC-BCS转变中的涨落效应对转化的影响,并找到稳定玻色子存在的参数区域,其标志为解离温度大于相变温度.
We investigate the crossover from Bardeen-Cooper-Schrieffer(BCS)pairing to a Bose-Einstein condensate(BEC)in a relativistic superfluid within a boson-fermion model.The model includes,besides the fermions,separate bosonic degrees of free-dom,accounting for the bosonic nature of the Cooper pairs.The crossover is realized by tuning the difference between the boson mass and boson chemical potential or directly the bare boson mass as a free parameter.As a mean field approxima-tion(MFA),we neglect the coupling between the thermal bosons with the fermions, the model yields populations of condensed and uncondensed bosons as well as gapped and ungapped fermions throughout the crossover region for arbitrary temperatures. Moreover,we observe the appearance of antiparticles for sufficiently large values of the crossover parameter.As an application,we study pairing of fermions with im-balanced populations and find the disappearing of the stable solution in the strong coupling region.
To go beyond the MFA we use Cornwall-Jackiw-Tomboulis(CJT)formalism to include higher order contributions.The quantum fluctuations of the pairing condensate is provided by bosons in non-zero modes,whose interaction with fermions gives the two-particle-irreducible(2PI)effective potential.It changes the crossover property in the BEC regime.We calculate the condensate,the critical temperature T_c and particle abundances as functions of crossover parameter the squared boson mass.We also compute the dissociation temperature T* for bosons,above which they are not stable and start to decay.We find the crossover parameter range for T~*＞T_c,which indicates the existence of bosons even above T_c but below T~*.
The model can potentially be applied to color superconductivity in dense quark matter at strong couplings.Or it can be extended to discuss the quarkoyinic conti-nuity.
The main innovations in this thesis are listed below:
●For the first time,we investigate the crossover from BCS pairing to a BEC in a relativistic superfluid within a boson-fermion model.
●In mean field approximation,the model yields populations of condensed and uncondensed bosons as well as gapped and ungapped fermions throughout the crossover.
●We observe the appearance of antiparticles for sufficiently large coupling as a characteristic of relativistic model.
●We study the pairing of fermions with imbalanced populations in BCS-BEC crossover,and find the disappearing of the stable solution in the strong coupling region.
●We also study the fluctuation effect in the crossover and find the parameter region for stable bosons where T~*＞T_c.
引文
[1]H.K.Onnes,Comm.Phys.Lab.Univ.Leiden,Nos.119,120,122(1911).
[2]W.Meissner and R.Ochsenfeld,Naturwiss.21,787(1933).
[3]J.Bardeen,L.N.Cooper,and J.R.Schrieffer,Phys.Rev.108,1175(1957).
[4]D.M.Eagles,Phys.Rev.186,456(1969);V[5]A.J.Leggett,in Modern Trends in the Theory of Condensed Matter,SpringerVerlag (1980);
[6]P.Nozieres and S.Schmitt-Rink,J.Low.Temp.Phys.59,195(1985).
[7]Y.Nishida and H.Abuki,Phys.Rev.D 72,096004(2005)[arXiv:hepph/0504083].
[8]C.A.Regal,M.Greiner,and D.S.Jin,Phys.Rev.Lett.92,040403(2004)[arXiv:cond-mat/0401554];M.Bartenstein et.al.,Phys.Rev.Lett.92,120401(2004)[arXiv:cond-mat/0401109];M.W.Zwierlein et.al.,Phys.Rev.Lett.92,120403(2004)[arXiv:cond-mat/0403049];J.Kinast et.al.,Phys.Rev.Lett.92,150402(2004)[arXiv:cond-mat/0403540];T.Bourdel et.al.,Phys.Rev.Lett.93,050401(2004)[arXiv:cond-mat/0403091].
[9]R..Friedberg and T.D.Lee,Phys.Rev.B 40,6745(1989);R.Friedberg,T.D.Lee and H.C.Ren,Phys.Rev.B 42,4122(1990).
[10]J.Ranninger in Bose-Einstein condensation,edited by A.Griffin,D.W.Snoke,and S.Stringari(Cambridge University Press,Cambridge,1995);T.Domanski,J.Ranninger,Physica C 387,77(2003)[arXiv:cond-mat/0208255].
[11]J.C.Collins and M.J.Perry,Phys.Rev.Lett.34,1353(1975).
[12]B.C.Barrois,Nucl.Phys.B 129,390(1977);
[13]D.Bailin and A.Love,Phys.Rept.107,325(1984).
[14]For reviews,see M.G.Alford,K.Rajagopal and F.Wilczek,Nucl.Phys.B 537,443(1999)[arXiv:hep-ph/9804403].K.Rajagopal and F.Wilczek,arXiv:hepph/0011333;S.Reddy,Acta Phys.Polon.B 33,4101(2002)[arXiv:nuclth/0211045];D.H.Rischke,Prog.Part.Nucl.Phys.52,197(2004)[arXiv:nuclth/0305030];M.Alford,Prog.Theor.Phys.Suppl.153,1(2004)[arXiv:nuclth/0312007];H.c.Ren,arXiv:hep-ph/0404074;M.Huang,Int.J.Mod.Phys.E 14,675(2005)[arXiv:hep-ph/0409167];I.A.Shovkovy,Found.Phys.35,1309(2005)[arXiv:nucl-th/0410091];T.Schafer,arXiv:hep-ph/0509068.
[15]See e.e.M.Gyulassy and L.Mclerran,Nucl.Phys.A 750,30(2005)
[16]Q.Chen,J.Stajic,S.Tan,K.Levin,Phys.Rep.412,1(2005).
[17]K.Nawa,E.Nakano and H.Yabu,Phys.Rev.D 74,034017(2006)[arXiv:hepph/0509029];A.H.Rezaeian and H.J.Pirner,Nucl.Phys.A 779,197(2006)[arXiv:nucl-th/0606043].
[18]M.Kitazawa,D.H.Rischke and I.A.Shovkovy,Phys.Lett.B 637,367(2006)[arXiv:hep-ph/0602065].
[19]H.Abuki,arXiv:hep-ph/0605081.
[20]L.He and P.Zhuang,arXiv:hep-ph/0703042.
[21]M.Grether,M.de Llano and G.A..Baker,Phys.Rev.Lett.99,200406(2007)[arXiv:0706.2833[cond-mat.supr-con]].
[22]M.G.Alford,K.Rajagopal and F.Wilczek,Phys.Lett.B 422,247(1998)[arXiv:hep-ph/9711395].
[23]R.Rapp,T.Schafer,E.V.Shuryak and M.Velkovsky,Phys.Rev.Lett.81,53(1998)[arXiv:hep-ph/9711396].
[24]S.B.Ruster,V.Werth,M.Buballa,I.A.Shovkovy and D.H.Rischke,Phys.Rev.D 72,034004(2005)[arXiv:hep-ph/0503184].
[25]I.A.Shovkovy and M.Huang,Phys.Lett.B564,205(2003).
[26]M.Alford,C.Kouvaris and K.Rajagopal,Phys.Rev.Lett.92,222001(2004)
[27]M.Huang and I.A.Shovkovy,Phys.Rev.D 70,051501(2004)[arXiv:hepph/0407049];
[28]R.Casalbuoni,R.Gatto,M.Mannarelli,G.Nardulli and M.Ruggieri,Phys.Lett.B 605,362(2005)[Erratum-ibid.B 615,297(2005)][arXiv:hep-ph/0410401].
[29]I.Giannakis,H.-C.Ren,Phys.Lett.B611,137(2005);I.Giannakis,D.-F.Hou,H.-C.Ren,Phys.Lett.B631,16(2005)
[30]K.Rajagopal and A.Schmitt,Phys.Rev.D 73,045003(2006)[arXiv:hepph/0512043].
[31]T.Schafer,[arXiv:hep-ph/0508190]
[32]E.V.Gorbar,M.Hashimoto and V.A.Miranshy,Phys.Lett.B632,305(2006)
[33]A.Schmitt,Q.Wang and D.Rischke,Phys.Rev.Lett91,242301(2003)
[34]D.T.Son,Phys.Rev.D 59,094019(1999)[arXiv:hep-ph/9812287];
[35]R.D.Pisarski and D.H.Rischke,Phys.Rev.D 61,074017(2000)[arXiv:nuclth/9910056].
[36]For recent developments in color superconductivity using perturbative QCD,see Q.Wang and D.H.Rischke,Phys.Rev.D 65,054005(2002)[arXiv:nuclth/0110016];A.Schmitt,Q.Wang and D.H.Rischke,Phys.Rev.D 66,114010(2002)[arXiv:nucl-th/0209050];A.Ipp,A.Gerhold and A.Rebhan,Phys.Rev.D 69,011901(2004)[arXiv:hep-ph/0309019];Q.Wang,J.Phys.G 30,S1251(2004)[arXiv:nucl-th/0404017];P.T.Reuter,Q.Wang and D.H.Rischke,Phys.Rev.D 70,114029(2004)[Erratum-ibid.D 71,099901(2005)][arXiv:nuclth/0405079];A.Gerhold and A.Rebhan,Phys.Rev.D 71,085010(2005)[arXiv:hep-ph/0501089];P.T.Reuter,arXiv:nucl-th/0602043;J.L.Noronha,H.c.Ren,I.Giannakis,D.Hou and D.H.Rischke,Phys.Rev.D 73,094009(2006)[arXiv:hep-ph/0602218];D.Nickel,J.Wambach and R.Alkofer,Phys.Rev.D 73,114028(2006)[arXiv:hep-ph/0603163];B.Feng,D.f.Hou,J.r.Li and H.c.Ren,Nucl.Phys.B 754,351(2006)[arXiv:nucl-th/0606015];P.T.Reuter,Phys.Rev.D 74,105008(2006)[arXiv:nucl-th/0608020].
[37]A.Gerhold and A.Rebhan,Phys.Rev.D68,011502(2003)
[38]D.-F.Hou,Q.Wang and D.Rischke,Phys.Rev.D69,071501(2004)
[39]D.K.Hong,Phys.Lett.B473,118(2000)
[40]T.Schafer,Nucl.Phys.A715,879(2003)
[41]P.T.Reuter,Q.Wang and D.H.Rischke,Phys.Rev.D 70,114029(2004)[Erratum-ibid.D 71,099901(2005)][arXiv:nucl-th/0405079].
[42]C.A.Regal,M.Greiner,and D.S.Jin,Phys.Rev.Lett.92,040403(2004)[arXiv:cond-mat/0401554].
[43]M.Bartenstein et.al.,Phys.Rev.Lett.92,120401(2004)[arXiv:condmat/0401109].
[44]M.W.Zwierlein et.al.,Phys.Rev.Lett.92,120403(2004)[arXiv:condmat/0403049].
[45]J.Kinast et.al.,Phys.Rev.Lett.92,150402(2004)[arXiv:cond-mat/0403540].
[46]T.Bourdel et.al.,Phys.Rev.Lett.93,050401(2004)[arXiv:cond-mat/0403091].
[47]C.Chin et.al.,Scinece 305,1128(2004)
[48]Nobel lectures:http://nobelprize.org/nobel_prizes/physics/laureates/2004/wilczek-lecture.html;http://nobelprize.org/nobel_prizes/physics/laureates/2004/gross-lecture.html;
[49]Report of the workshop on Bev/necleon collisions of heavy ions - how and why,Bear Mountain,New York,Nov.29 - Dec.1,1974(BNL-AUI,1975)
[50]T.D.Lee and G.C.Wick,Phys.Rev.D 9,2291(1974)
[51]J.Hofmann,H.Stocker,W.Scheid,W.Greiner,Bear Mountain,New York,Nov.29- Dec.1,1974(BNL-AUI,1975)
[52]www.gsi.de/experiments/CBM/index_e.html.
[53]http://chandra.harvard.edu /photo/2002/0211/
[54]D.T.Son and M.A.Stephanov,Phys.Rev.Lett.86,592(2001)[arXiv:hepph/0005225];D.T.Son and M.A.Stephanov,Phys.Atom.Nucl.64,834(2001)[Yad.Fiz.64,899(2001)][arXiv:hep-ph/0011365];L.y.He,M.Jin and P.f.Zhuang,Phys.Rev.D 71,116001(2005)[arXiv:hep-ph/0503272];G.f.Sun,L.He and P.Zhuang,arXiv:hep-ph/0703159.
[55]S.B.Ruester,V.Werth,M.Buballa,I.A.Shovkovy and D.H.Rischke,Phys.Rev.D 72,034004(2005)[arXiv:hep-ph/0503184].
[56]M.Kitazawa,D.H.Rischke and I.A.Shovkovy,Prog.Theor.Phys.Suppl.168,389(2007)[arXiv:0707.3966[nucl-th]].
[57]M.Kitazawa,D.H.Rischke and I.A.Shovkovy,arXiv:0709.2235[hep-ph].
[58]L.P.Kandanoff and P.C.Martin,Phys.Rev.123(3),670
[59]L.He and P.Zhuang,Phys.Rev.D 76,056003(2007)[arXiv:0705.1634[hep-ph]].
[60]M.Holland,S.J.J.M.F.Kokkelmans,M.L.Chiofalo,R.Walser,Phys.Rev.Lett.87,120406(2001)[arXiv:cond-mat/0103479];M.L.Chiofalo,S.J.J.M.F.Kokkelmans,J.N.Milstein,M.J.Holland,Phys.Rev.Lett.88,090402(2001)[arXiv:cond-mat/0110119];E.Timmermans,K.Furuya,P.W.Milonni,A.K.Kerman,Phys.Lett.A 285,228.(2001)[arXiv:cond-mat/0103327];Y.Ohashi and A.Griffin,Phys.Rev.Lett.89,130402(2002).
[61]J.Deng,A.Schmitt and Q.Wang,Phys.Rev.D 76,034013(2007)[arXiv:nuclth /0611097].
[62]J.Deng,J.C.Wang and Q.Wang,[arXiv:0803.4360](hep-ph)
[63]Y.Nambu and G.Jona-Lasinio,Phys.Rev.122 345-358(1961),Phys.Rev.124246-254(1961).
[64]M.Buballa,Phys.Rept.407,205(2005)[arXiv:hep-ph/0402234].
[65]T.Hatsuda and T.Kunihiro,Phys.Rept.247,221(1994)[arXiv:hepph/9401310].
[66]J.I.Kapusta,Finite-temperature field theory(Cambridge University Press,Cambridge,1989).
[67]M.Alford and Q.h.Wang,J.Phys.G 31,719(2005)[arXiv:hep-ph/0501078].
[68]M.Kitazawa,T.Koide,T.Kunihiro and Y.Nemoto,Prog.Theor.Phys.114,117(2005)[arXiv:hep-ph/0502035];M.Kitazawa,T.Kunihiro and Y.Nemoto,Phys.Lett.B 631,157(2005)[arXiv:hep-ph/0505070].
[69]S.Y.Chang,J.Carlson,V.R.Pandharipande,K.E.Schmidt,Phys.Rev.A.70,043602(2004)[arXiv:physics/0404115].
[70]J.Carlson and S.Reddy,Phys.Rev.Lett.95,060401(2005)[arXiv:condmat/0503256].
[71]M.W.Zwierlein,A.Schirotzek,C.H.Schunck and W.Ketterle,Science 311,492(2006)[arXiv:cond-mat/0511197];G.B.Partridge,W.Li,R.I.Kamar,Y.Liao and R.G.Hulet,Science 311,503(2006)[arXiv:cond-mat/0511752].
[72]Y.Nishida and D.T.Son,Phys.Rev.Lett.97,050403(2006)[arXiv:condmat/0604500];A.Bulgac and M.M.Forbes,arXiv:cond-mat/0606043;G.Rupak,T.Schafer and A.Kryjevski,arXiv:cond-mat/0607834;Y.Nishida and D.T.Son,arXiv:cond-mat/0607835;J.-W.Chen,E.Nakano,arXiv:cond-mat/0610011.
[73]E.Gubankova,W.V.Liu and F.Wilczek,Phys.Rev.Lett.91,032001(2003)[arXiv:hep-ph/0304016].
[74]C.-H.Pao,S.-T.Wu,S.-K.Yip,Phys.Rev.B 73,132506(2006);ibid,74,189901(E)(2006)[arXiv:cond-mat/0506437];D.T.Son and M.A.Stephanov,Phys.Rev.A 74,013614(2006)[arXiv:cond-mat/0507586];M.Mannarelli,G.Nardulli and M.Ruggieri,Phys.Rev.A 74,033606(2006)[arXiv:cond-mat/0604579];Q.Chen,Y.He,C.-C.Chien,K.Levin,Phys.Rev.A 74,063603(2006)[arXiv:condmat/0608454];D.E.Sheehy and L.Radzihovsky,Phys.Rev.B 75,136501(2007).
[75]E.Gubankova,A.Schmitt and F.Wilczek,Phys.Rev.B 74,064505(2006)[arXiv:cond-mat/0603603].
[76]A.I.Larkin and Yu.N.Ovchinnikov,Zh.Eksp.Teor.Fiz.47,1136(1964)[Sov.Phys.JETP 20,762(1965)];P.Fulde and R.A.Ferrell,Phys.Rev.135,A550(1964).
[77]I.Giannakis,D.f.Hou,H.c.Ren and D.H.Rischke,Phys.Rev.Lett.93,232301(2004)[arXiv:hep-ph/0406031].
[78]J.M.Cornwall,R.Jackiw and E.Tomboulis,Phys.Rev.D 10,2428(1974).
[79]T.Brauner,arXiv:0803.2422(hep-ph).
[80]H.s.Zong,L.Chang,F.y.Hou,W.m.Sun and Y.x.Liu,Phys.Rev.C 71,015205(2005).L.Chang,Y.X.Liu,M.S.Bhagwat,C.D.Roberts and S.V.Wright,Phys.Rev.C 75,015201(2007)[arXiv:nucl-th/0605058].
[81]D.Nickel,J.Wambach and R.Alkofer,Phys.Rev.D 73,114028(2006)[arXiv:hep-ph/0603163].
[82]F.Marhauser,D.Nickel,M.Buballa and J.Wambach,Phys.Rev.D 75,054022(2007)[arXiv:hep-ph/0612027].
[83]M.Kitazawa,T.Koide,T.Kunihiro and Y.Nemoto,Phys.Rev.D 65,091504(2002)[axXiv:nucl-th/0111022].
[84]M.Kitazawa,T.Koide,T.Kunihiro and Y.Nemoto,Phys.Rev.D 70,056003(2004)[axXiv:hep-ph/0309026].
[2]W.Meissner and R.Ochsenfeld,Naturwiss.21,787(1933).
[3]J.Bardeen,L.N.Cooper,and J.R.Schrieffer,Phys.Rev.108,1175(1957).
[4]D.M.Eagles,Phys.Rev.186,456(1969);V[5]A.J.Leggett,in Modern Trends in the Theory of Condensed Matter,SpringerVerlag (1980);
[6]P.Nozieres and S.Schmitt-Rink,J.Low.Temp.Phys.59,195(1985).
[7]Y.Nishida and H.Abuki,Phys.Rev.D 72,096004(2005)[arXiv:hepph/0504083].
[8]C.A.Regal,M.Greiner,and D.S.Jin,Phys.Rev.Lett.92,040403(2004)[arXiv:cond-mat/0401554];M.Bartenstein et.al.,Phys.Rev.Lett.92,120401(2004)[arXiv:cond-mat/0401109];M.W.Zwierlein et.al.,Phys.Rev.Lett.92,120403(2004)[arXiv:cond-mat/0403049];J.Kinast et.al.,Phys.Rev.Lett.92,150402(2004)[arXiv:cond-mat/0403540];T.Bourdel et.al.,Phys.Rev.Lett.93,050401(2004)[arXiv:cond-mat/0403091].
[9]R..Friedberg and T.D.Lee,Phys.Rev.B 40,6745(1989);R.Friedberg,T.D.Lee and H.C.Ren,Phys.Rev.B 42,4122(1990).
[10]J.Ranninger in Bose-Einstein condensation,edited by A.Griffin,D.W.Snoke,and S.Stringari(Cambridge University Press,Cambridge,1995);T.Domanski,J.Ranninger,Physica C 387,77(2003)[arXiv:cond-mat/0208255].
[11]J.C.Collins and M.J.Perry,Phys.Rev.Lett.34,1353(1975).
[12]B.C.Barrois,Nucl.Phys.B 129,390(1977);
[13]D.Bailin and A.Love,Phys.Rept.107,325(1984).
[14]For reviews,see M.G.Alford,K.Rajagopal and F.Wilczek,Nucl.Phys.B 537,443(1999)[arXiv:hep-ph/9804403].K.Rajagopal and F.Wilczek,arXiv:hepph/0011333;S.Reddy,Acta Phys.Polon.B 33,4101(2002)[arXiv:nuclth/0211045];D.H.Rischke,Prog.Part.Nucl.Phys.52,197(2004)[arXiv:nuclth/0305030];M.Alford,Prog.Theor.Phys.Suppl.153,1(2004)[arXiv:nuclth/0312007];H.c.Ren,arXiv:hep-ph/0404074;M.Huang,Int.J.Mod.Phys.E 14,675(2005)[arXiv:hep-ph/0409167];I.A.Shovkovy,Found.Phys.35,1309(2005)[arXiv:nucl-th/0410091];T.Schafer,arXiv:hep-ph/0509068.
[15]See e.e.M.Gyulassy and L.Mclerran,Nucl.Phys.A 750,30(2005)
[16]Q.Chen,J.Stajic,S.Tan,K.Levin,Phys.Rep.412,1(2005).
[17]K.Nawa,E.Nakano and H.Yabu,Phys.Rev.D 74,034017(2006)[arXiv:hepph/0509029];A.H.Rezaeian and H.J.Pirner,Nucl.Phys.A 779,197(2006)[arXiv:nucl-th/0606043].
[18]M.Kitazawa,D.H.Rischke and I.A.Shovkovy,Phys.Lett.B 637,367(2006)[arXiv:hep-ph/0602065].
[19]H.Abuki,arXiv:hep-ph/0605081.
[20]L.He and P.Zhuang,arXiv:hep-ph/0703042.
[21]M.Grether,M.de Llano and G.A..Baker,Phys.Rev.Lett.99,200406(2007)[arXiv:0706.2833[cond-mat.supr-con]].
[22]M.G.Alford,K.Rajagopal and F.Wilczek,Phys.Lett.B 422,247(1998)[arXiv:hep-ph/9711395].
[23]R.Rapp,T.Schafer,E.V.Shuryak and M.Velkovsky,Phys.Rev.Lett.81,53(1998)[arXiv:hep-ph/9711396].
[24]S.B.Ruster,V.Werth,M.Buballa,I.A.Shovkovy and D.H.Rischke,Phys.Rev.D 72,034004(2005)[arXiv:hep-ph/0503184].
[25]I.A.Shovkovy and M.Huang,Phys.Lett.B564,205(2003).
[26]M.Alford,C.Kouvaris and K.Rajagopal,Phys.Rev.Lett.92,222001(2004)
[27]M.Huang and I.A.Shovkovy,Phys.Rev.D 70,051501(2004)[arXiv:hepph/0407049];
[28]R.Casalbuoni,R.Gatto,M.Mannarelli,G.Nardulli and M.Ruggieri,Phys.Lett.B 605,362(2005)[Erratum-ibid.B 615,297(2005)][arXiv:hep-ph/0410401].
[29]I.Giannakis,H.-C.Ren,Phys.Lett.B611,137(2005);I.Giannakis,D.-F.Hou,H.-C.Ren,Phys.Lett.B631,16(2005)
[30]K.Rajagopal and A.Schmitt,Phys.Rev.D 73,045003(2006)[arXiv:hepph/0512043].
[31]T.Schafer,[arXiv:hep-ph/0508190]
[32]E.V.Gorbar,M.Hashimoto and V.A.Miranshy,Phys.Lett.B632,305(2006)
[33]A.Schmitt,Q.Wang and D.Rischke,Phys.Rev.Lett91,242301(2003)
[34]D.T.Son,Phys.Rev.D 59,094019(1999)[arXiv:hep-ph/9812287];
[35]R.D.Pisarski and D.H.Rischke,Phys.Rev.D 61,074017(2000)[arXiv:nuclth/9910056].
[36]For recent developments in color superconductivity using perturbative QCD,see Q.Wang and D.H.Rischke,Phys.Rev.D 65,054005(2002)[arXiv:nuclth/0110016];A.Schmitt,Q.Wang and D.H.Rischke,Phys.Rev.D 66,114010(2002)[arXiv:nucl-th/0209050];A.Ipp,A.Gerhold and A.Rebhan,Phys.Rev.D 69,011901(2004)[arXiv:hep-ph/0309019];Q.Wang,J.Phys.G 30,S1251(2004)[arXiv:nucl-th/0404017];P.T.Reuter,Q.Wang and D.H.Rischke,Phys.Rev.D 70,114029(2004)[Erratum-ibid.D 71,099901(2005)][arXiv:nuclth/0405079];A.Gerhold and A.Rebhan,Phys.Rev.D 71,085010(2005)[arXiv:hep-ph/0501089];P.T.Reuter,arXiv:nucl-th/0602043;J.L.Noronha,H.c.Ren,I.Giannakis,D.Hou and D.H.Rischke,Phys.Rev.D 73,094009(2006)[arXiv:hep-ph/0602218];D.Nickel,J.Wambach and R.Alkofer,Phys.Rev.D 73,114028(2006)[arXiv:hep-ph/0603163];B.Feng,D.f.Hou,J.r.Li and H.c.Ren,Nucl.Phys.B 754,351(2006)[arXiv:nucl-th/0606015];P.T.Reuter,Phys.Rev.D 74,105008(2006)[arXiv:nucl-th/0608020].
[37]A.Gerhold and A.Rebhan,Phys.Rev.D68,011502(2003)
[38]D.-F.Hou,Q.Wang and D.Rischke,Phys.Rev.D69,071501(2004)
[39]D.K.Hong,Phys.Lett.B473,118(2000)
[40]T.Schafer,Nucl.Phys.A715,879(2003)
[41]P.T.Reuter,Q.Wang and D.H.Rischke,Phys.Rev.D 70,114029(2004)[Erratum-ibid.D 71,099901(2005)][arXiv:nucl-th/0405079].
[42]C.A.Regal,M.Greiner,and D.S.Jin,Phys.Rev.Lett.92,040403(2004)[arXiv:cond-mat/0401554].
[43]M.Bartenstein et.al.,Phys.Rev.Lett.92,120401(2004)[arXiv:condmat/0401109].
[44]M.W.Zwierlein et.al.,Phys.Rev.Lett.92,120403(2004)[arXiv:condmat/0403049].
[45]J.Kinast et.al.,Phys.Rev.Lett.92,150402(2004)[arXiv:cond-mat/0403540].
[46]T.Bourdel et.al.,Phys.Rev.Lett.93,050401(2004)[arXiv:cond-mat/0403091].
[47]C.Chin et.al.,Scinece 305,1128(2004)
[48]Nobel lectures:http://nobelprize.org/nobel_prizes/physics/laureates/2004/wilczek-lecture.html;http://nobelprize.org/nobel_prizes/physics/laureates/2004/gross-lecture.html;
[49]Report of the workshop on Bev/necleon collisions of heavy ions - how and why,Bear Mountain,New York,Nov.29 - Dec.1,1974(BNL-AUI,1975)
[50]T.D.Lee and G.C.Wick,Phys.Rev.D 9,2291(1974)
[51]J.Hofmann,H.Stocker,W.Scheid,W.Greiner,Bear Mountain,New York,Nov.29- Dec.1,1974(BNL-AUI,1975)
[52]www.gsi.de/experiments/CBM/index_e.html.
[53]http://chandra.harvard.edu /photo/2002/0211/
[54]D.T.Son and M.A.Stephanov,Phys.Rev.Lett.86,592(2001)[arXiv:hepph/0005225];D.T.Son and M.A.Stephanov,Phys.Atom.Nucl.64,834(2001)[Yad.Fiz.64,899(2001)][arXiv:hep-ph/0011365];L.y.He,M.Jin and P.f.Zhuang,Phys.Rev.D 71,116001(2005)[arXiv:hep-ph/0503272];G.f.Sun,L.He and P.Zhuang,arXiv:hep-ph/0703159.
[55]S.B.Ruester,V.Werth,M.Buballa,I.A.Shovkovy and D.H.Rischke,Phys.Rev.D 72,034004(2005)[arXiv:hep-ph/0503184].
[56]M.Kitazawa,D.H.Rischke and I.A.Shovkovy,Prog.Theor.Phys.Suppl.168,389(2007)[arXiv:0707.3966[nucl-th]].
[57]M.Kitazawa,D.H.Rischke and I.A.Shovkovy,arXiv:0709.2235[hep-ph].
[58]L.P.Kandanoff and P.C.Martin,Phys.Rev.123(3),670
[59]L.He and P.Zhuang,Phys.Rev.D 76,056003(2007)[arXiv:0705.1634[hep-ph]].
[60]M.Holland,S.J.J.M.F.Kokkelmans,M.L.Chiofalo,R.Walser,Phys.Rev.Lett.87,120406(2001)[arXiv:cond-mat/0103479];M.L.Chiofalo,S.J.J.M.F.Kokkelmans,J.N.Milstein,M.J.Holland,Phys.Rev.Lett.88,090402(2001)[arXiv:cond-mat/0110119];E.Timmermans,K.Furuya,P.W.Milonni,A.K.Kerman,Phys.Lett.A 285,228.(2001)[arXiv:cond-mat/0103327];Y.Ohashi and A.Griffin,Phys.Rev.Lett.89,130402(2002).
[61]J.Deng,A.Schmitt and Q.Wang,Phys.Rev.D 76,034013(2007)[arXiv:nuclth /0611097].
[62]J.Deng,J.C.Wang and Q.Wang,[arXiv:0803.4360](hep-ph)
[63]Y.Nambu and G.Jona-Lasinio,Phys.Rev.122 345-358(1961),Phys.Rev.124246-254(1961).
[64]M.Buballa,Phys.Rept.407,205(2005)[arXiv:hep-ph/0402234].
[65]T.Hatsuda and T.Kunihiro,Phys.Rept.247,221(1994)[arXiv:hepph/9401310].
[66]J.I.Kapusta,Finite-temperature field theory(Cambridge University Press,Cambridge,1989).
[67]M.Alford and Q.h.Wang,J.Phys.G 31,719(2005)[arXiv:hep-ph/0501078].
[68]M.Kitazawa,T.Koide,T.Kunihiro and Y.Nemoto,Prog.Theor.Phys.114,117(2005)[arXiv:hep-ph/0502035];M.Kitazawa,T.Kunihiro and Y.Nemoto,Phys.Lett.B 631,157(2005)[arXiv:hep-ph/0505070].
[69]S.Y.Chang,J.Carlson,V.R.Pandharipande,K.E.Schmidt,Phys.Rev.A.70,043602(2004)[arXiv:physics/0404115].
[70]J.Carlson and S.Reddy,Phys.Rev.Lett.95,060401(2005)[arXiv:condmat/0503256].
[71]M.W.Zwierlein,A.Schirotzek,C.H.Schunck and W.Ketterle,Science 311,492(2006)[arXiv:cond-mat/0511197];G.B.Partridge,W.Li,R.I.Kamar,Y.Liao and R.G.Hulet,Science 311,503(2006)[arXiv:cond-mat/0511752].
[72]Y.Nishida and D.T.Son,Phys.Rev.Lett.97,050403(2006)[arXiv:condmat/0604500];A.Bulgac and M.M.Forbes,arXiv:cond-mat/0606043;G.Rupak,T.Schafer and A.Kryjevski,arXiv:cond-mat/0607834;Y.Nishida and D.T.Son,arXiv:cond-mat/0607835;J.-W.Chen,E.Nakano,arXiv:cond-mat/0610011.
[73]E.Gubankova,W.V.Liu and F.Wilczek,Phys.Rev.Lett.91,032001(2003)[arXiv:hep-ph/0304016].
[74]C.-H.Pao,S.-T.Wu,S.-K.Yip,Phys.Rev.B 73,132506(2006);ibid,74,189901(E)(2006)[arXiv:cond-mat/0506437];D.T.Son and M.A.Stephanov,Phys.Rev.A 74,013614(2006)[arXiv:cond-mat/0507586];M.Mannarelli,G.Nardulli and M.Ruggieri,Phys.Rev.A 74,033606(2006)[arXiv:cond-mat/0604579];Q.Chen,Y.He,C.-C.Chien,K.Levin,Phys.Rev.A 74,063603(2006)[arXiv:condmat/0608454];D.E.Sheehy and L.Radzihovsky,Phys.Rev.B 75,136501(2007).
[75]E.Gubankova,A.Schmitt and F.Wilczek,Phys.Rev.B 74,064505(2006)[arXiv:cond-mat/0603603].
[76]A.I.Larkin and Yu.N.Ovchinnikov,Zh.Eksp.Teor.Fiz.47,1136(1964)[Sov.Phys.JETP 20,762(1965)];P.Fulde and R.A.Ferrell,Phys.Rev.135,A550(1964).
[77]I.Giannakis,D.f.Hou,H.c.Ren and D.H.Rischke,Phys.Rev.Lett.93,232301(2004)[arXiv:hep-ph/0406031].
[78]J.M.Cornwall,R.Jackiw and E.Tomboulis,Phys.Rev.D 10,2428(1974).
[79]T.Brauner,arXiv:0803.2422(hep-ph).
[80]H.s.Zong,L.Chang,F.y.Hou,W.m.Sun and Y.x.Liu,Phys.Rev.C 71,015205(2005).L.Chang,Y.X.Liu,M.S.Bhagwat,C.D.Roberts and S.V.Wright,Phys.Rev.C 75,015201(2007)[arXiv:nucl-th/0605058].
[81]D.Nickel,J.Wambach and R.Alkofer,Phys.Rev.D 73,114028(2006)[arXiv:hep-ph/0603163].
[82]F.Marhauser,D.Nickel,M.Buballa and J.Wambach,Phys.Rev.D 75,054022(2007)[arXiv:hep-ph/0612027].
[83]M.Kitazawa,T.Koide,T.Kunihiro and Y.Nemoto,Phys.Rev.D 65,091504(2002)[axXiv:nucl-th/0111022].
[84]M.Kitazawa,T.Koide,T.Kunihiro and Y.Nemoto,Phys.Rev.D 70,056003(2004)[axXiv:hep-ph/0309026].